So here's what you get if, for each cell in a grid, you sample n from the Poisson distribution, and put in n circular splotches of value, where the center of the circle is uniformly distributed in that square grid cell.
And then do that all over again for various sizes of grid and circle.
Since the Poisson process is isotropic, and the distribution of number of events in each region is Poisson-distributed and independent from all other regions, the result is 100% definitely isotropic.
http://jcreed.org/js/roads/continents.html